Quasi-Monte Carlo — Discrepancy between Theory and Practice

  • Shu Tezuka


In this paper, we survey the current status of theory and practice of quasi-Monte Carlo methods. First, we discuss one of the most important research directions for accelerating Monte Carlo simulations. It is described as MC → QMC → RQMC → Derandomized RQMC, where RQMC means randomized QMC. We give some interesting open questions concerning the gap between theory and practice of derandomized RQMC. Secondly, we overview the dramatic success of quasi-Monte Carlo methods for very high dimensional numerical integration problems in finance. In the last five years, the question of how to explain this success has been extensively investigated, and two classes of problems have been identified for which QMC (or RQMC) is much more efficient than MC. One is a class of problems with small “effective” dimensions; the other is a class of isotropic problems. Some interesting results and issues on these problems are discussed.


Effective Dimension Integration Error Lattice Rule Mortgage Back Security Isotropic Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Atanassov, E. I. (March 2000) On the Discrepancy of the Halton Sequences. manuscriptGoogle Scholar
  2. 2.
    Braaten, E., Weiler, G. (1979) An Improved Low Discrepancy Sequence for Multidimensional Quasi-Monte Carlo Integration. J. Comput. Phys., 3.3, 249–258CrossRefGoogle Scholar
  3. 3.
    Caflisch, R. E., Morokoff, W., Owen, A. (1997) Valuation of Mortgage Backed Securities using Brownian Bridges to Reduce Effective Dimension. Journal of Computational Finance, 1, 27–46Google Scholar
  4. 4.
    Case, J. (December 1995) Wall Street's Dalliance with Number Theory. SIAM News, 28, 10, 8–9Google Scholar
  5. 5.
    Chen, W. W. L., Skriganov, M. M. (2000) Explicit Constructions in the Classical Mean Squares Problem in Irregularities of Point Distribution, manuscriptGoogle Scholar
  6. 6.
    Cranley, R., Patterson, T. (1976) Randomization of Number Theoretic Methods for Multiple Integration. SIAM Journal of Numerical Analysis, 13, 904–914MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Curbera, F. (2001) Algorithms and Complexity for Gaussian Integration and Applications to Financial Problems, PhD Thesis, Columbia University, New YorkGoogle Scholar
  8. 8.
    Faure, H. (1982) Discrepances de suites associées a un système de numeration en dimension s. Acta Arith., XLI, 337–351MathSciNetGoogle Scholar
  9. 9.
    Faure, H. (1992) Good Permutation for Extreme Discrepancy. J. Number Theory, 41, 47–56MathSciNetCrossRefGoogle Scholar
  10. 10.
    Faure, H., Tezuka, S. (2001) A New Generation of Digital (0, 5)-Sequences. Research Report RT0412, IBM Tokyo Research LaboratoryGoogle Scholar
  11. 11.
    Fox, B. L. (1999) Strategies for Quasi-Monte Carlo, Kluwer Academic Publishers, BostonCrossRefGoogle Scholar
  12. 12.
    Hickernell F. J. (2000) What Affects the Accuracy of Quasi-Monte Carlo Quadrature? in Monte Carlo and Quasi-Monte Carlo Methods 1998, edited by H. Niederreit er and J. Spanier, Springer, Berlin, 16–55Google Scholar
  13. 13.
    Matousek, J. (1999) Geometric Discrepancy: An Illustrated Guide, Springer, BerlinMATHCrossRefGoogle Scholar
  14. 14.
    Matousek, J. (1998) On the L 2-Discrepancy for Anchored Boxes. Journal of Complexity, 14, 527–556MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Niederreiter, H. (1978) Quasi-Monte Carlo Methods and Pseudorandom Numbers. Bull. Amer. Math. Soc, 84, 957–1041MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Niederreiter, H. (1987) Point Sets and Sequences with Small Discrepancy. Monatsh. Math., 104, 273–337MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Niederreiter, H. (1992) Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Regional Conference Series in Applied Mathematics, No.63, SIAMGoogle Scholar
  18. 18.
    Niederreiter, H., Xing, C. (1998) Nets, (t,s)-sequences, and Algebraic Geometry, in Random and Quasi-Random Point Sets, edited by P. Hellekalek and G. Larcher, Lecture Notes in Statistics, 138, Springer, New York, 267–302Google Scholar
  19. 19.
    Novak, E., Ritter, K. (1996) High Dimensional Integration of Smooth Functions over Cubes. Numer. Math., 75, 79–97MathSciNetMATHGoogle Scholar
  20. 20.
    Novak, E., Woźniakowski, H. (2001) When are Integration and Discrepancy Tractable? in Foundations of Computational Mathematics, edited by R.A. DeVore et al., London Mathematical Society Lecture Note Series 284, Cambridge Univ. Press, 211–266Google Scholar
  21. 21.
    Owen, A. (1997a) Scrambled Net Variance for Integrals of Smooth Functions. The Annals of Statistics, 25(4), 1541–1562MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Owen, A. (1997b) How Nearly Linear is a Function? Technical Report, Statistics Dept., Stanford UniversityGoogle Scholar
  23. 23.
    Owen, A. (2000) Monte Carlo, Quasi-Monte Carlo, and Randomized Quasi-Monte Carlo. in Monte Carlo and Quasi-Monte Carlo Methods 1998, edited by H. Niederreiter and J. Spanier, Springer, Berlin, 86–97Google Scholar
  24. 24.
    Papageorgiou, A. (2001) Fast Convergence of Quasi-Monte Carlo for a Class of Isotropic Integrals. Math. Comp., 70, 233, 297–306MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Papageorgiou, A., Traub, J. F. (June 1996) Beating Monte Carlo. RISK, 9, 63–65Google Scholar
  26. 26.
    Papageorgiou, A., Traub, J. F. (Nov/Dec 1997) Faster Evaluation of Multidimensional Integrals. Computers in Physics, 11, 574–578CrossRefGoogle Scholar
  27. 27.
    Paskov, S. H. (1997) New Methodologies for Valuing Derivatives. in Mathematics of Derivative Securities, edited by M. A. H. Dempster and S. Pliska, Isaac Newton Institute, Cambridge University Press, Cambridge UK, 545–582Google Scholar
  28. 28.
    Paskov, S. H., Traub, J. F. (Fall 1995) Faster Valuation of Financial Derivatives, Journal of Portfolio Management, 22(1), 113–120CrossRefGoogle Scholar
  29. 29.
    Sloan, I. H. (2001) QMC Integration — Beating Intractability by Weighting the Coordinate Directions, in this volume Google Scholar
  30. 30.
    Sloan, I. H., Kuo, F. Y., Joe, S. (October 2000) On the Step-by-Step Construction of Quasi-Monte Carlo Integration Rules That Achieve Strong Tractability Error Bounds in Weighted Sobolev Spaces. Applied Math. Report AMR00/24, University of New South WalesGoogle Scholar
  31. 31.
    Sloan, I. H., Woźniakowski, H. (1998) When are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals. Journal of Complexity, 14, 1–33MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Tezuka, S. (1995) Uniform Random Numbers: Theory and Practice, Kluwer Academic Publishers, BostonMATHCrossRefGoogle Scholar
  33. 33.
    Tezuka, S. (1998) Financial Applications of Monte Carlo and Quasi-Monte Carlo Methods, in Random and Quasi-Random Point Sets, edited by P. Hellekalek and G. Larcher, Lecture Notes in Statistics, 138, Springer, New York, 303–332Google Scholar
  34. 34.
    Tezuka, S. (2000) Quasi-Monte Carlo Methods for Financial Applications, in ICIAM99, edited by J. M. Ball and J. C. R. Hunt, Oxford Univ. Press, New York, 234–245Google Scholar
  35. 35.
    Tezuka, S. (2000) Discrepancy Theory and Its Application to Finance, in IFIP TCS2000, edited by J. van Leeuwen et al., Lecture Notes in Computer Science, 1872, Springer, New York, 243–256Google Scholar
  36. 36.
    Traub, J. F., Werschulz, A. G. (1998) Complexity and Information, Cambridge Univ. PressGoogle Scholar
  37. 37.
    Warnock, T. (1972) Computational Investigation of Low-Discrepancy Point Sets. in Applications of Number Theory to Numerical Analysis, edited by S. K. Zaremba, Academic Press, New York, 319–343Google Scholar
  38. 38.
    Woźniakowski, H. (1991) Average Case Complexity of Multivariate Integration. Bull. Amer. Math. Soc., 24, 185–194MathSciNetMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Shu Tezuka
    • 1
  1. 1.IBM Tokyo Research LaboratoryYamato, KanagawaJapan

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